Progression of Diabetic Capillary Occlusion: A Model
Fig 5
Procedure of sequential simulation of advection and diffusion illustrated with an example.
During each time step, simulation of advection precedes simulation of other processes including diffusion. After advection (involving the CBs delivering the blood containing oxygen) and before diffusion (involving CAPs), a CAP always sums up oxygen volumes in all its associated CBs and update its pre-diffusion oxygen volume. Simulation of diffusion updates the CAP to have its post-diffusion oxygen volume. After diffusion and before advection at the next time step, the associated CBs must have the same relative change of oxygen volumes as the CAP gains or loses during diffusion, namely, percent change from pre-diffusion to post-diffusion values. This mathematically recognizes the conservation of oxygen. Three consecutive time steps are shown in the example. A naïve assumption here for this illustration is that the first CB always receives 1 unit oxygen volume from upstream (not drawn). Another simplification made for this illustration is that only diffusion between the CAP (shown in red) and nearby OT (shown in light brown) are considered. In the actual model OT to OT diffusion is also treated. Note that in a certain step, boldface numbers represent values being changed or updated. During the first time step from t0 to t0 + Δt, while the second CAP and its associated CBs still have a zero oxygen volume, the first CAP and associated CBs undergo (1) the process of advection that passes 1 unit oxygen volume to first CB, while CAPs are not involved; (2) an intermediate step that updates CAP’s pre-diffusion oxygen volume by adding 1 (its first associated CB) and 0 (its second associated CB); (3) the process of diffusion delivers 0.2 to OT in contact (amount assumed for convenience in this example, and again OT-OT diffusion is ignored in this example) and CAP’s post-diffusion oxygen volume becomes 0.8, while CBs are not involved; (4) a last step in the time period that updates CAP’s associated CBs’ oxygen volumes by subtracting 0.2/1 = 20% (diffused/pre-diffusion), the first CB thus having 0.8 oxygen volume now. During the second time step from t0 + Δt to t0 + 2Δt, similar verbal “simulation” goes. (1) process of advection goes as another 1 oxygen volume is passed to first CB and 0.8, previously held by the first CB, is passed to the second CB; (2) an intermediate step adds 1 and 0.8 to first CAP, still none added to the second CAP; (3) the process of diffusion updates first CAP’s oxygen volume to 1.44, with 0.36 diffused out; (4) last step updates oxygen volumes in both of the host CAP’s two CBs, again by subtracting diffused fraction 0.36/1.8 = 20%. During the third time step from t0 + 2Δt to t0 + 3Δt, advection now passes oxygen volume 0.64, previously held by the second CB, into the third CB, which is associated with the second CAP. An intermediate step updates both CAPs by summing up oxygen volumes in their associated CBs. Process of diffusion now changes the oxygen volumes of both CAPs, with the first and second diffusing out 0.36/1.8 = 20% and 0.1/0.64 = 15.625% respectively. The last step subtracts the oxygen volumes of their asscociated CBs’ with the percent change.